for some suitable even function \(f\) . We want to analyze the spectral action for perturbations \(D\rightarrow D+V \)  by bosonic gauge fields of the form \(V =a_j [D ,b_j]\)  (summation over \(j\)  understood). Even though our analysis is valid in the general setting of noncommutative geometry [1]}, [2]} the most interesting cases that occur in physics are:
